Let $(x_{i,j})$ be a double sequence of nonnegative real numbers, and $ 0< p<1$.

I would like to know whether one can bound from above the sum \begin{equation} \sum_{i,j} x_{i,j}^p \end{equation} in terms of \begin{equation} \sum_{i}\Big(\sum_{j} x_{i,j}\Big)^p \quad \text{and} \quad \sum_{j}\Big(\sum_{i} x_{i,j}\Big)^p \quad ? \end{equation} The bound does not have to be tight, any upper bound will do.

My first guess was \begin{equation} \sum_{i,j} x_{i,j}^p \leq \sum_{i}\Big(\sum_{j} x_{i,j}\Big)^p + \sum_{j}\Big(\sum_{i} x_{i,j}\Big)^p + \sum_{i}\Big(\sum_{j} x_{i,j}\Big)^p \cdot \sum_{j}\Big(\sum_{i} x_{i,j}\Big)^p \end{equation} but I was unsuccessful in proving it so far.